Friday, February 20, 2015

Is Quaternionic Quantum Mechanics Detectable?

Let us start by explaining why quaternionic quantum mechanics may be interesting for describing nature despite its problem with the tensor product.

There are several arguments put forward, but I think only two of them have merit. The first obvious usage can be in justifying the \(SU(2)\) gauge group of the Standard Model which can be achieved in a similar fashion of how \(U(1)\) (the gauge group of electromagnetism) arises when one considers local symmetry.

The second potential usage stems from the peculiar behavior of quaternionic quantum mechanics: the ground energy level is uniquely determined and the zero point energy cannot be freely shifted like in complex quantum mechanics. The obvious application of this is in solving the puzzle of the cosmological constant which by naive field theory arguments should be \({10}^{120}\) times larger than the observed value.

A pure state in quaternionic quantum mechanics is defined only up to a quaternionic phase (a unit quaternion) and one may ask if such phases which are different than complex wavefunction phases are experimentally detectable. In fact Asher Peres.

Asher Peres

proposed such an experiment: pass a neutron beam through slabs of dissimilar materials and search for the non-commutativity of the phase shift when the slabs are reversed. The experiment was performed and the answer was negative. The theoretical clarification came later on from Adler's analysis which showed that the S-matrix quaternionic scattering is in fact indistinguishable from the usual complex quantum mechanics scattering.

This result is not surprising given the previous post: quaternionic quantum mechanics can be understood as a constrained complex quantum mechanics and there the square roof of negative one is represents a map between observables (hermitean operators) and generators of continuous symmetries (anti-hermitean operators). This map is also known as "dynamic correspondence".

Before comparing quaternionic quantum mechanics with real and complex quantum mechanics there is one last result of relative importance. Wigner's theorem states that in complex quantum mechanics symmetries can be represented by a unitary or anti-unitary transformation. Emch, Piron, Uhlhorn and Bargman generalized this to the quaternionic quantum mechanics case and here there are only quaternionic-unitary transformations.